Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #153664 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13694 |
| commutator, Frattini, cosocle | C2×C32 | C23 |
| prime radical | #41 | D16 |
| LCS | G ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #131 ◁ #504 ◁ #2258 ◁ G | |
| LECS | G ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
110 conjugacy classes of subgroups.
| minimal degree | μ(G) = 256 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 256 points:
(1,2,10,18)(3,19,27,78)(4,28,33,65)(5,34,39,124)(6,17,9,14)(7,43,46,136)(8,47,48,67)(11,49,55,22)(12,40,59,30)(13,60,64,20)(15,68,70,35)(16,42,71,44)(21,85,90,175)(23,77,26,74)(24,98,101,209)(25,102,103,177)(29,108,112,216)(31,115,116,221)(32,117,118,58)(36,123,38,120)(37,129,130,224)(41,133,45,135)(50,142,145,194)(51,92,54,94)(52,149,151,184)(53,152,153,183)(56,154,156,186)(57,157,158,217)(61,81,63,84)(62,163,164,189)(66,125,69,126)(72,95,171,87)(73,140,174,79)(75,139,179,91)(76,97,180,99)(80,105,159,161)(82,122,128,225)(83,121,160,162)(86,104,138,146)(88,198,199,254)(89,200,201,170)(93,137,141,148)(96,206,100,208)(106,167,165,109)(107,132,166,114)(110,119,127,213)(111,218,219,223)(113,134,131,220)(143,233,234,256)(144,195,222,229)(147,237,150,239)(155,188,214,241)(168,231,243,248)(169,230,244,252)(172,182,173,185)(176,202,178,203)(181,190,228,245)(187,215,226,255)(191,227,246,193)(192,205,247,197)(196,207,204,253)(210,238,235,250)(211,232,236,249)(212,240,242,251);(1,3,20,79,7,24,82,183,44,99,189,239,126,203,38,94,9,26,84,185,45,100,130,141,48,103,160,149,68,139,39,49,10,27,60,140,46,101,128,152,42,97,163,237,125,202,36,92,6,23,81,182,41,96,37,93,8,25,83,184,35,91,5,22)(2,11,34,75,15,52,121,102,47,137,129,206,133,173,63,77,17,54,123,178,69,150,164,180,71,153,225,209,136,174,64,78,18,55,124,179,70,151,162,177,67,148,224,208,135,172,61,74,14,51,120,176,66,147,62,76,16,53,122,98,43,73,13,19)(4,21,80,181,31,88,187,249,114,197,251,256,217,252,110,194,30,87,186,248,113,196,219,229,118,201,241,238,167,227,112,138,33,90,159,228,116,199,226,232,132,205,240,233,157,230,127,142,40,95,154,231,131,204,111,195,32,89,188,250,109,193,29,86)(12,50,119,169,57,143,212,192,107,211,215,198,115,190,105,85,28,104,108,191,106,210,214,200,117,222,218,253,220,243,156,171,59,145,213,244,158,234,242,247,166,236,255,254,221,245,161,175,65,146,216,246,165,235,155,170,58,144,223,207,134,168,56,72);(1,4)(2,12)(3,21)(5,29)(6,40)(7,31)(8,32)(9,30)(10,33)(11,50)(13,56)(14,65)(15,57)(16,58)(17,28)(18,59)(19,72)(20,80)(22,86)(23,95)(24,88)(25,89)(26,87)(27,90)(34,119)(35,109)(36,127)(37,111)(38,110)(39,112)(41,131)(42,132)(43,134)(44,114)(45,113)(46,116)(47,107)(48,118)(49,138)(51,146)(52,143)(53,144)(54,104)(55,145)(60,159)(61,161)(62,155)(63,105)(64,156)(66,165)(67,166)(68,167)(69,106)(70,158)(71,117)(73,168)(74,175)(75,169)(76,170)(77,85)(78,171)(79,181)(81,154)(82,187)(83,188)(84,186)(91,193)(92,142)(93,195)(94,194)(96,204)(97,205)(98,207)(99,197)(100,196)(101,199)(102,192)(103,201)(108,123)(115,133)(120,216)(121,212)(122,223)(124,213)(125,157)(126,217)(128,226)(129,215)(130,219)(135,221)(136,220)(137,211)(139,227)(140,228)(141,229)(147,235)(148,236)(149,238)(150,210)(151,234)(152,232)(153,222)(160,241)(162,242)(163,240)(164,214)(172,245)(173,190)(174,243)(176,246)(177,247)(178,191)(179,244)(180,200)(182,231)(183,249)(184,250)(185,248)(189,251)(198,206)(202,230)(203,252)(208,254)(209,253)(218,225)(224,255)(233,237)(239,256)Polycyclic presentation:
2^9|a^2=i,b^2=df,d^2=fg,e^2=i,f^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=df,e^a=ei,f^a=fg,g^a=gh,h^a=hi,e^c=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60861);